Proposition 6

Let E be a locally convex space, F a separated locally convex space, $u: E\rightarrow F$ a compact linear mapping and B a bounded subset of E. The restriction of $u$ to B is a continuous mapping of the set B, endowed with the topology induced by $\sigma (E,E')$, into the space F.

The mapping $u$ is a continuous linear mapping of E into F, and also of $E_{\sigma}$ into $F_{\sigma}$. Its restriction to B is therefore a continuous mapping of the set B, endowed with the topology induced by $\sigma (E,E')$, into the space $F_{\sigma}$. Now the set $u(B)$ is contained in a compact subset C of F (remark 1 of III, p. 2) and the space $F_{\sigma}$ is separated, so that the topologies induced on C by those of F and of $F_{\sigma}$ coincide. The proposition follows.